Classifications

• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L27/00—Modulated-carrier systems
  • H04L27/0004—Modulated-carrier systems using wavelets
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L27/00—Modulated-carrier systems
  • H04L27/26—Systems using multi-frequency codes
  • H04L27/2601—Multicarrier modulation systems
  • H04L27/2602—Signal structure
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L27/00—Modulated-carrier systems
  • H04L27/26—Systems using multi-frequency codes
  • H04L27/2601—Multicarrier modulation systems
  • H04L27/2602—Signal structure
  • H04L27/26025—Numerology, i.e. varying one or more of symbol duration, subcarrier spacing, Fourier transform size, sampling rate or down-clocking
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L27/00—Modulated-carrier systems
  • H04L27/26—Systems using multi-frequency codes
  • H04L27/2601—Multicarrier modulation systems
  • H04L27/2626—Arrangements specific to the transmitter only
  • H04L27/2627—Modulators
  • H04L27/264—Pulse-shaped multi-carrier, i.e. not using rectangular window
  • H04L27/26416—Filtering per subcarrier, e.g. filterbank multicarrier [FBMC]
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L27/00—Modulated-carrier systems
  • H04L27/26—Systems using multi-frequency codes
  • H04L27/2601—Multicarrier modulation systems
  • H04L27/2647—Arrangements specific to the receiver only
  • H04L27/2649—Demodulators
  • H04L27/26534—Pulse-shaped multi-carrier, i.e. not using rectangular window
  • H04L27/2654—Filtering per subcarrier, e.g. filterbank multicarrier [FBMC]
• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L27/00—Modulated-carrier systems
  • H04L27/26—Systems using multi-frequency codes
  • H04L27/2601—Multicarrier modulation systems
  • H04L27/2697—Multicarrier modulation systems in combination with other modulation techniques
  • H04L27/2698—Multicarrier modulation systems in combination with other modulation techniques double density OFDM/OQAM system, e.g. OFDM/OQAM-IOTA system

Definitions

• the field of the invention is that of the transmission or dissemination of digital data, or of analog and sampled data, intended to be received in particular by mobiles. More specifically, the invention relates to signals produced using new modulations, as well as the corresponding modulation and demodulation techniques.
• the OFDM was selected as part of this European project as the basis for the DAB standard. This technique is also envisaged as modulation for the broadcasting of television programs.
• This technique is also envisaged as modulation for the broadcasting of television programs.
• the invention finds applications in many fields, in particular when high spectral efficiency is desired and the channel is highly non-stationary.
• a first category of applications concerns digital terrestrial broadcasting, be it image, sound and / or data.
• the invention can be applied to synchronous broadcasting, which intrinsically generates multiple paths of long duration. It also advantageously applies to broadcasting to mobiles.
• the invention can find applications in particular in digital communication systems to high speed mobiles, in the context for example of UMTS (RACE project). It can also be considered for local radio networks broadband (HIPERLAN type).
• a third category of applications is that of underwater transmissions.
• the underwater acoustic transmission channel is highly disturbed due to the low transmission speed of the acoustic waves in the water. This leads to a large spread of the multiple paths and of the Doppler spectrum.
• Multicarrier modulation techniques are therefore well suited to this field, and very particularly the techniques which are the subject of the present invention.
• F A denotes the analytical filter
• the signal s (t) belongs to a vector subspace (characterized by the band limitation to the space of the complex functions of a real variable and of square
• COFDM This coding technique with interleaving in the time-frequency plane is known as COFDM. It is for example described in document [23] (see appendix 1 (to simplify the reading, most of the references of the state of the art are listed in this appendix 1. This one, as well as appendices 2 and 3 must of course be considered as full-fledged elements of this description)).
• a first category of modulations consists of a multiplex of QAM (Quadrature Amplitude Modulation) carriers, or possibly in QPSK (Quadrature Phase Shift Keying) in the particular case of binary data.
• QAM Quadrature Amplitude Modulation
• QPSK Quadrature Phase Shift Keying
• the coefficients a mn take complex values representing the data transmitted.
• the functions x m, n (t) are translated in time-frequency space of the same prototype function x (t):
• ⁇ being any phase, which we can arbitrarily set to 0.
• the function x (t) is centered, that is to say that its moments of order 1 are zero, that is:
• the barycenters of the basic functions therefore form a network of the time-frequency plane generated by the vectors ( ⁇ 0 , 0) and (0, v 0 ), as illustrated in FIG. 1.
• the prototype function x (t) has the particularity that the functions ⁇ x m, n ⁇ are orthogonal to each other, and more precisely constitute a Hilbert basis of L 2 (R), that is:
• Projecting a signal under this base is simply equivalent to cutting the signal into sequences of duration ⁇ 0 and to representing each of these sequences by the corresponding development in Fourier series.
• This type of decomposition constitutes a first step towards a localization both in time and in frequency, in contrast to the classic Fourier analysis, which ensures a perfect frequency localization with a total loss of temporal information.
• the parameter ⁇ is 0 for OFDM / QAM. This is a major defect in the modulation
• OFDM / QAM OFDM / QAM as described above. This is characterized in practice by a high sensitivity to time errors, and therefore to multiple paths.
• This fault can be overcome by using a guard interval described for example in [5].
• This is a device consisting in extending the rectangular window of the prototype function. The density of the network of basic symbols is then strictly less than unity.
• OFDM / QAM modulation with guard interval is the basis of the system
• This guard interval makes it possible to limit the interference between symbols, at the cost of a loss of performance, since part of the information transmitted is not actually used by the receiver, but only serves to absorb the multiple paths.
• the loss is 1 dB.
• there is an additional loss due to the fact that to obtain a given overall spectral efficiency, it is necessary to compensate for the loss due to the guard interval by a better efficiency of the code used.
• the case of the rectangular window corresponds to OFDM / QAM without guard interval.
• the case of the cardinal sinus corresponds to a conventional frequency multiplex (that is to say of which the carriers have disjoint spectra) with a roll-off of
• OFDM / QAM corresponding to the use of a density 1 network and to complex am.n coefficients, can only be put into practice in the case of a rectangular time window and the use a guard interval.
• OFDM / OQAM Those skilled in the art looking for other modulations are therefore led to turn to the techniques described below under the name OFDM / OQAM.
• a second category of modulations indeed uses a multiplex of OQAM carriers (Offset Quadrature Amplitude Modulation).
• OFDM / OQAM Offset Quadrature Amplitude Modulation
• the carriers are all synchronized, and the carrier frequencies are spaced half the inverse of the symbol time. Although the spectra of these carriers overlap, the synchronization of the system and the choice of the phases of the carriers makes it possible to guarantee orthogonality between the symbols emitted by different carriers.
• References [11-18] give a good overview of the literature available on this subject. For simplicity in writing, we will represent the signals in their analytical form. Under these conditions, the general equation of an OFDM / OQAM signal is written:
• the coefficients a m, n take real values representing the data transmitted.
• the functions x mjn (t) are translated in time-frequency space of the same prototype function x (t):
• ⁇ being any phase which one can arbitrarily fixed equal to 0.
• the barycentres of the basic functions therefore form a network of the time-frequency plane generated by the vectors ( ⁇ 0 , 0) and (0, v 0 ), as illustrated in FIG. 2.
• This network is of density 2.
• the functions x m, n (t) are orthogonal in the sense of the dot product in R.
• the prototype function is bounded in frequency, so that the spectrum of each carrier does not overlap than that of the adjacent carriers.
• the prototype functions considered are even (real or possibly complex) even functions satisfying the following relationship:
• x (t) is the impulse response of a 100% roll-off half-Nyquist filter, i.e.
• the frequency performance of the OFDM / OQAM is rather satisfactory and the problem of loss linked to the guard interval does not arise.
• the impulse response of the prototype function has a relatively slow temporal decrease, ie in 1 / x 2 .
• the waveform can hardly be truncated over a short time interval, which implies complex processing at the level of the receiver. In addition, this also complicates possible equalization systems.
• the invention particularly aims to overcome these various drawbacks and limitations of the state of the art.
• an objective of the invention is to provide a digital signal intended to be transmitted or broadcast to receivers, which makes it possible to obtain better performance in non-stationary channels, and very particularly in strongly non-stationary channels.
• the invention also aims to provide such a signal, making it possible to obtain high spectral efficiency.
• Another objective of the invention is to provide such a signal, which avoids the drawbacks of OFDM / QAM linked to the guard interval, while keeping a temporal response of the prototype function as concentrated as possible, in particular so to simplify processing at the receiver.
• the invention also aims to provide such a signal, allowing the realization of receivers with limited complexity and cost, in particular with regard to demodulation and equalization.
• a complementary objective of the invention is to provide transmitters, receivers, transmission or broadcasting methods, reception methods and methods of construction, that is to say of definition, of a modulation corresponding to such a signal.
• a multicarrier signal intended to be transmitted to digital receivers, in particular in a non-stationary transmission channel, corresponding to the frequency multiplexing of several elementary carriers each corresponding to a series of symbols, two consecutive symbols being separated by a symbol time ⁇ 0 , a signal in which, on the one hand, the spacing v 0 between two neighboring carriers is equal to half the inverse of time symbol ⁇ 0 ,
• each carrier undergoes shaping filtering of its spectrum having a bandwidth strictly greater than twice said spacing between carriers v 0 .
• This spectrum is chosen so that each symbol element is concentrated as much as possible both in the time domain and in the frequency domain.
• such a signal can respond to the following equation:
• n is a real coefficient representative of the source signal, chosen from a predetermined modulation alphabet;
• m is an integer representing the frequency dimension;
• n is an integer representing the time dimension
• t represents time
• x m, n (t) is a basic function, translated in time-frequency space of the same prototype function x (t) pair taking real or complex values, that is:
• ⁇ is an arbitrary phase parameter
• ⁇ indicates that x m, n (t) can take either a negative or a positive sign. It does not mean, of course, that x m, n (t) takes both values.
• the invention is based on a modulation system which uses prototype functions as concentrated as possible in the time-frequency plane.
• the interest of this approach is to have a modulation producing a signal avoiding the disadvantages of OFDM / QAM related to the guard interval, while keeping a temporal response of the prototype function as concentrated as possible, so to simplify processing at the receiver.
• the subject of the invention is new modulation systems constructed like OFDM / OQAM on an orthogonal network of density 2, without however the prototype function being of frequency bounded support.
• the proposed modulations there are either modulations using prototype functions with time bound support, or prototype functions which are bound neither in time nor in frequency, but which exhibit rapid decay properties at the same time and in frequency, and a quasi-optimal concentration in the time-frequency plane.
• the first known construction method uses a density network 1.
• This first solution uses a signal decomposition base where any signal is cut into intervals, each interval then being broken down into Fourier series. It is the OFDM / QAM solution.
• the literature gives few examples of alternative solutions built on the same network, and the results obtained are of little practical interest [10].
• the OFDM / QAM technique is the only one that can benefit from the guard interval method.
• the OFDM / QAM solution is therefore a singular point which does not allow extensions.
• the second known construction method uses a density 2 network.
• the orthogonality between symbols centered on the same frequency or on adjacent frequencies is guaranteed by the formatting of half-type signals.
• said prototype function x (t) is an even function, zero outside the interval [- ⁇ 0 , ⁇ 0 ], and verifying the relation:
• said prototype function x (t) is defined by:
• OFDM / MSK the performances in terms of resistance to Doppler and to multiple paths are equivalent to OFDM / OQAM, and the construction of the receiver is simplified.
• said prototype function x (t) is characterized by the equation: the function y (t) being defined by its Fourier transform Y (f):
• the parameter a is equal to unity.
• the corresponding modulation is hereinafter called OFDM / IOTA.
• the corresponding prototype function, denoted 3 is identical to its Fourier transform.
• the invention also relates to a method for transmitting a digital signal, in particular in a non-stationary transmission channel, comprising the following steps:
• such a method further comprises a frequency and / or time interleaving step, applied to the binary elements forming said digital signal to be transmitted or to the digital coefficients a m, n .
• the invention also relates to the transmitters of such a signal.
• the invention also relates to a method of receiving a signal as described above, which comprises the following steps:
• this reception method comprises a step of deinterlacing in frequency and / or in time of said real digital coefficients â m, n and, optionally, of the corresponding values ⁇ m, n of the response of the amplitude of the channel, said deinterlacing being inverse of an interleaving implemented on transmission, and / or a step of decoding in weighted decision adapted to the channel coding implemented on transmission.
• the invention also relates to the corresponding receivers.
• the invention also relates to a preferred method of constructing a prototype function x (t) for a signal as described above, comprising the following steps:
• G (f) (2a) 1/4 e -raf 2 ;
• FIG. 1 illustrates a density network 1, corresponding to that used in the case of the known OFDM / QAM modulation
• FIG. 2 illustrates a density network 2, corresponding to that used in the case of the known OFDM / OQAM modulation and in the case of the invention
• FIG. 3A to 3D, 4C to 4D, 5A to 5D, 6A to 6D and 7A to 7D respectively illustrate the known modulations OFDM / QAM (3), OFDM / QAM with guard interval (4), OFDM / OQAM (5 ) and the modulations of the invention OFDM / MSK (6) and OFDM / IOTA (7), according to the following aspects:
• . B the linear Fourier transform of the prototype function
• . C the module of the linear ambiguity function (as defined in appendix 2);
• FIG. 9 is a block diagram of a transmitter (and the corresponding transmission method) usable according to the invention.
• FIG. 10 is a block diagram of a receiver (and the corresponding reception method) usable according to the invention.
• FIG. 11 illustrates more precisely the demodulation method implemented in the receiver of FIG. 10.
• the barycentres of the basic functions therefore form a network of the time-frequency plane generated by the vectors (r 0 , 0) and (0, v 0 ) (see Figure 2).
• these functions constitute a Hilbert base of H R. In order to simplify the writing, we will later omit the sign inversions.
• Orthogonality is therefore obtained if the coefficient of the integral is a pure imaginary number. Analysis of this coefficient shows that it is sufficient for this that m-m 'or that n-n' is odd.
• the network can therefore be broken down into four subnetworks, as shown in Figure 2 ( ⁇ m even, n even ⁇ , ⁇ m even, n odd ⁇ , ⁇ m odd n even ⁇ , ⁇ m odd, n odd ⁇ ) which are orthogonal to each other (any function of one of the subnets is orthogonal to any function of another subnetwork).
• the invention is based on a completely new approach to multicarrier signals of the OFDM / OQAM type, according to which the orthogonality is no longer obtained by respecting the two constraints mentioned above, but by a specific definition of the prototype functions.
• the subject of the invention is new signals, based on modulation systems constructed such as OFDM / OQAM on an orthogonal network of density 2, without however the prototype function being of frequency bounded support. .
• the principle used is to construct orthogonal networks of density 1/2, then to deduce from them networks of density 2 by a judicious choice of the phases of the signals.
• this modulation can be considered as dual of the OFDM / OQAM, since it corresponds to an exchange of the time and frequency axes.
• the main advantage of this modulation compared to OFDM / OQAM is that the prototype function is strictly limited in time, which considerably simplifies the implementation of the receiver, since the number of coefficients of the input filter is considerably reduced . Furthermore, the performance in the presence of multiple paths is unchanged, the parameter ⁇ being identical.
• . B the linear Fourier transform of the prototype function
• . C the module of the linear ambiguity function (as defined in appendix 2);
• D the intersymbol function (as defined in appendix 2).
• the views of the ambiguity function make it possible to judge the confinement in the time-frequency plane of the prototype function.
• the views of the intersymbol function (indexed figures D) make it possible to appreciate the sensitivity of a modulation to the delay and to the Doppler. Phase errors are not considered, since all the modulations are equivalent on this plane.
• FIGS. 3A to 3D relate to the known case of conventional OFDM / QAM.
• the main defect in this modulation is not, as the frequency response of the prototype function might suggest, the slight decrease in the level of the secondary lobes.
• the real problem comes from the brutal limitation of the temporal response, which corresponds to a triangular ambiguity function along this axis. This gives an intersymbol function with a very high sensitivity to temporal errors: the slope is vertical and the parameter ⁇ is therefore zero. This is what justifies the use of a guard interval.
• Figures 4C and 4D relate to OFDM / QAM with a guard interval (the prototype and Fourier transform functions are identical to those of OFDM / QAM illustrated in Figures 3A and 3C.
• the use of a guard interval creates a flat area at the level of the ambiguity function. In fact, we should rather speak in this case of cross-ambiguity. We obviously find this flat part at the level of the intersymbol function, which gives immunity to errors
• the figures represent the case of a guard interval 0.25 ⁇ 0 .
• the cost of the guard interval is admissible when one is interested in modulations with low spectral efficiency. On the other hand, it becomes prohibitive if we seek a high spectral efficiency: let us take for example a guard interval equal to a quarter of the useful symbol. Under these conditions, to achieve a net efficiency of 4 bits / s / Hertz, a modulation and coding system having a raw efficiency of 5 bits / s / Hertz, ie a loss of 3 dB compared to the limit capacity of Shannon. It is also necessary to add to this loss the additional loss of 1 dB due to the power "unnecessarily" emitted in the guard interval. In total, therefore, 4 dB is lost compared to the optimum.
• Figures 5A to 5D present the case of OFDM / OQAM.
• the time response of OFDM / OQAM has a better appearance than that of OFDM / QAM. However, the temporal decrease is only 1 / t 2 . The ambiguity function is canceled out on a 1/2 density network. The sensitivity to frequency errors is higher than that to time errors.
• the parameter ⁇ is worth 0.8765.
• FIGS. 6A to 6D relate to the first embodiment of the invention, the OFDM / MSK. We verify that it has properties strictly identical to those of OQAM by inverting the time and frequency scales. The parameter ⁇ is unchanged.
• FIGS. 7A to 7D show the OFDM / IOTA modulation. This presents a rapid decrease (in the mathematical sense of the term) in time and frequency, which makes it possible to envisage equalization in the best possible conditions. It also has perfect symmetry with respect to these two axes. Its intersymbol function is almost ideal. In general, her behavior is similar to that of the Gaussian. The parameter ⁇ is worth 0.9769.
• FIG. 7E shows on a logarithmic scale the decrease in time of the IOTA signal.
• the amplitude of the signal decreases linearly on a logarithmic scale (in time and in frequency, of course, since the two aspects are identical), that is to say exponentially in linear scale. This property therefore makes it possible in a practical embodiment to truncate the waveform and thus limit the complexity of the receiver.
• FIG. 9 shows a simplified block diagram of a signal transmitter according to the invention. The emission process is directly deduced therefrom.
• binary source is meant a series of data elements corresponding to one or more source signals 91 of all types (sounds, images, data) sampled digital or analog.
• This binary data is subjected to a binary to binary channel coding 92 adapted to vanishing channels.
• trellis code Trellis Coded Modulation
• Reed-Solomon More specifically, if you want a spectral efficiency of 4 bits / Hertz, you can use a 2/3 efficiency code associated with an 8AM modulation, taking 8 amplitude levels.
• a first binary to binary coding is carried out, a time and frequency interleaving and a binary coding with coefficients, commonly called “mapping”. It is clear that the interleaving can be carried out either before or after the mapping, depending on the needs and the codes used.
• This operation can advantageously be carried out in digital form by a fast Fourier transform (FFT) relating to all the symbols of the same rank n, followed by a multiplication of the resulting waveform by the prototype function IOTA, and finally by an addition of the symbols of different ranks (summation according to the index n).
• FFT fast Fourier transform
• the complex signal thus generated is then converted into analog form 98, then transposed to the final frequency by a modulator 99 with two quadrature channels.
• FIG. 10 schematically illustrates a receiver of a signal according to the invention (as well as the corresponding reception method).
• the OFDM / MSK or OFDM / IOTA receiver is substantially similar to that suitable for OFDM / OQAM modulation.
• the entrance floors are conventional.
• the signal is preamplified 101, then converted to intermediate frequency 102 in order to perform channel filtering 103.
• the signal to intermediate frequency is then converted to baseband on two quadrature channels 105.
• the automatic correction functions of gain (CAG) 104 which controls the preamplification 101.
• Another solution consists in transposing the signal into an intermediate frequency on a low carrier frequency, so as to sample the signal on a single channel, the complex representation then being obtained by digital filtering.
• the RF signal can be transposed directly into the baseband (direct conversion), the channel filtering then being performed on each of the two I&Q channels. In all cases, it can be reduced to a discrete representation of the signal of the complex envelope corresponding to the received signal.
• the demodulator estimates (106) the transfer function T (f, t) by conventional means, which can for example use a reference network of explicit carriers according to patent FR-9001491.
• the channel is locally assimilated to a multiplicative channel characterized by an amplitude and a phase corresponding to the value of T (f, t) for the time being and the frequency considered.
• the received signal is therefore assimilated to the signal:
• T (mv 0 , n ⁇ 0 ) ⁇ m, n e i ⁇ m, n (34)
• the receiver therefore performs the following processing:
• the processing 107 is carried out in digital form, according to the method illustrated in FIG. 11.
• the receiver operates in a similar fashion to an OFDM / OQAM receiver [13-16]. It performs the following treatments:
• correction 115 of the phase ⁇ m, n as a function of the estimate of the channel 116 comprising for example an estimate ⁇ m, n of the amplitude response and an estimate ⁇ m, n of the phase response of the transmission channel ;
• This algorithm therefore makes it possible to globally calculate all the coefficients of a given index n.
• the order of magnitude of the corresponding complexity is approximately double that of the algorithm used for OFDM / QAM.
• the coefficients thus obtained are then deinterleaved 108, symmetrically to the interleaving implemented on transmission, then decoded 109, advantageously according to a soft decision decoding technique, for example implementing an algorithm of the type of the algorithm of Viterbi. If the channel decoding takes account of the estimation of the response of the amplitude of the channel ⁇ m, n , the corresponding values are also deinterlaced 110. Furthermore, the deinterlacing is of course carried out before or after the mapping, depending on the time when interleaving was implemented on transmission.
• a dispersive channel can be considered as a linear system with a time-varying impulse response. There are two ways to define this impulse response. We will largely draw inspiration from the conventions proposed in [21]:
• V (v, ⁇ ) e i2 ⁇ vT U ( ⁇ , v)
• T (f, t) ⁇ g (t, ⁇ ) e -i2 ⁇ f ⁇ d ⁇
• T (f, t) ⁇ U ( ⁇ , v) e- i2 ⁇ t e i2 ⁇ vt d ⁇ dv
• the delay-Doppler model is defined by the equation:
• This equation shows the channel as a sum of elementary channels characterized by an amplitude, a phase, a time offset and a frequency offset.
• OFDM multi-carrier modulation of any type (OFDM / QAM, OFDM / OQAM or OFDM / IOTA) characterized by the generic equation:
• a k being real variables
• E being a two-dimensional network of density 2 in time-frequency space
• the functions x k (t) being time and frequency translates of the same prototype function x (t), and which constitute a Hilbert base of L 2 (R).
• ⁇ n Re [e -i ⁇ ⁇ r (t) x n * (t) dt]
• the second term represents Interference Between Symbols (IES). If we consider the data a k as independent random variables of variance ⁇ 2 , the variance I of the IES is written:
• a xn ( ⁇ , v) e 2i ⁇ (vn ⁇ - ⁇ nv) A x ( ⁇ , v)
• ⁇ opt ⁇ + ⁇ v ⁇ + 2 ⁇ ( ⁇ n vv n ⁇ )
• a x (d ⁇ , dv) 1 - 2 ⁇ 2 ( ⁇ t 2 dv 2 + ⁇ f 2 d ⁇ 2 ) + ⁇ dvd ⁇ + o (dv 2 + d ⁇ 2 )
• I (d ⁇ , dv, d ⁇ ) ⁇ 2 [4 ⁇ 2 ( ⁇ t 2 dv 2 + ⁇ f 2 d ⁇ 2 ) - 2 ⁇ dvd ⁇ + d ⁇ 2 + o (dv 2 + d ⁇ 2 + d ⁇ 2 )]
• This appendix gives a method of constructing prototype functions verifying the required orthogonality criteria.
• the method allows to obtain an infinity of functions, among which a particular solution (called IOTA function) having the particularity of being identical to its Fourier transform.
• a x ( ⁇ , v) A x ( ⁇ , -v)
• a xk ( ⁇ , v) e 2i ⁇ (vk ⁇ -v ⁇ k) ⁇ e -2i ⁇ vu x (u + ⁇ / 2)
• x * (u- ⁇ / 2) du e 2i ⁇ (vk ⁇ -v ⁇ k)
• the network ⁇ x m, n ⁇ can therefore be broken down into four sub-networks characterized by: ⁇ m even, n even ⁇ , ⁇ m even, n odd ⁇ , ⁇ m odd n even ⁇ , ⁇ m odd, n odd ⁇ .
• the orthogonality between functions belonging to different sub-networks is therefore automatic, and does not depend on the properties of the prototype function, from the moment it is even.
• ⁇ y (f, 2mv 0 ) c m ⁇ y (f, 0)
• a y ( ⁇ , 2mv 0 ) c m A y ( ⁇ , 0)
• the temporal orthogonalization operator O t therefore orthogonalizes the entire network, with the exception of the frequency axis.
• ⁇ z (t, 2n ⁇ 0 ) ⁇ y (t, 2n ⁇ 0 ) P (t)
• the ambiguity function of z vanishes outside of (0.0) for all the multiples of 2 ⁇ 0 and 2v 0 , i.e. a network of density 1/2.
• OF -1 OFx ⁇ F -1 OFOx a

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